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Cardy formula

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Cardy formula

In physics Cardy formula is important because it gives the entropy of black holes. Recent year, this formula has appeared in not only the calculation of the entropy of BTZ black holes but also the checking of the AdS/CFT correspondence and the holographic principle.

In 1986 J. L. Cardy discovered this formula Cardy (1986), which gives the entropy of (1+1)-dimensional conformal field theory (CFT)

S=2\pi\sqrt{\tfrac{c}{6}\bigl(L_0-\tfrac{c}{24}\bigr)},

where c is the central charge, L0 the product ER of the total energy and radius of system, and the shift of c/24 is caused by the Casimir effect. Here, c and L0 construct the Virasoro algebra of this CFT. In 2000 E. Verlinde extended this formula to the arbitrary (n+1)-dimensions Verlinde (2000), so it is also called Cardy-Verlinde formula. Consider a AdS space with the metric

ds^2=-dt^2+R^2\Omega^2_n

where R is the radius of a n-dimensional sphere. The dual CFT lives on the boundary of this AdS space. The entropy of the dual CFT can be given by this formula as

S=\frac{2\pi R}{n}\sqrt{E_c(2E-E_c)},

where Ec is the Casimir effect, E total energy. The above reduced formula gives the maximal entropy

S\le S_{max}=\frac{2\pi RE}{n},

when Ec=E. This is just the Bekenstein bound. The Cardy-Verlinde formula was later shown by Kutasov and Larsen to be invalid for weakly interacting CFTs. In fact, since the entropy of higher dimensional (meaning n>1) CFTs is dependent on exactly marginal couplings, it is believed that a Cardy formula for the entropy is not achievable when n>1. However, for supersymmetric CFTs, a twisted version of the partition function, called "the superconformal index" (related to the Witten index) is shown by Di Pietro and Komargodski to exhibit Cardy-like behavior when n=3 or 5.

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References

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